Cosine signal correction circuit

ABSTRACT

A cosine signal corrector circuit as for use in a decoder for a system wherein an information signal is transmitted in the form of one trigonometric transform function of that signal. The second trigonometric transform function signal, required for decoding the information signal, required for decoding the information signal, is obtained by first deriving the Hilbert transform of the transmitted signal and phase inverting it, producing an approximate second transform function signal. The decoder produces the sum of the squares of the two transform function signals, and the difference between that decoder output and 1.0 is an &#34;error&#34; signal. Under the control of the signum signal of the approximate second transform function signal, the error signal is added to the approximate second transform function signal to produce an accurate second transform function signal. Even if the first transform function signal and its Hilbert transform approach zero at the same time, the circuit provides the correct polarity for the error signal.

This application is a divisional application of the parent applicationSer. No. 165,474, filed July 2, 1980.

BACKGROUND OF THE INVENTION

This invention relates to the field of radio transmissions systemsproviding all of the advantages of present FM systems within one-halfthe normal bandwidth and, more particularly, to a cosine signalcorrection circuit for use in such a system.

It is well known that angular modulation systems have advantages overamplitude modulation systems, notably in the areas of signal-to-noiseratio and in the ability to "capture" a receiver. However, due to thesteadily increasing demand for more communication channels throughoutthe radio spectrum, the smaller band width required by AM-SSB systemshas begun to appear more desirable in spite of the considerabledisadvantages. Thus, there has developed a need for a system having theadvantages of FM within the bandwidth of the AM-SSB system.

Known angular modulation systems are of two main types. One system usesdirect frequency modulation of an oscillator which could not be a stableoscillator such as a crystal oscillator except with a very lowmodulation index. The usual phase modulated system uses a stableoscillator with narrow band modulation, then multiplies the frequency ofthe modulated signal to obtain the desired broadband signal.

SUMMARY OF THE INVENTION

It is therefore an object of the present invention to provide atransmission system having the advantages of FM transmission andapproximately one-half the bandwidth.

Another object is to provide an improved system of FM transmission. Aparticular object is to provide, in a receiver designed for such atransmission system, an improvement in the signal decoder circuit.

These objects and others which will become apparent are obtained in asystem wherein the modulating signal is one of the transform signals ofan FM system, with one set of sidebands removed. The second transformsignal can be recovered at the receiver to provide the original inputsignal. The accuracy of the second transform signal can be improved bythe use of an error-correcting signal which uses a form of feedback toforce the transform signal toward the exact value.

BRIEF DESCRIPTION OF THE DRAWING

FIG. 1 is a block diagram of an FM system.

FIGS. 2a, 2b and 2c are block diagrams of three embodiments of thedecoder of FIG. 1.

FIG. 3 is a block diagram of still another embodiment of the decoder ofFIG. 1.

FIG. 4a is a block diagram of the transform modulation system and FIG.4b is a chart of the corresponding waveforms.

FIG. 5 is a more detailed diagram relating to FIG. 4.

FIGS. 6a-6e are charts of waveforms related to FIGS. 5 and 8.

FIG. 7 is a block diagram of a portion of the diagram of FIG. 5.

FIG. 8 is a block diagram of a cosine signal correcting circuit to beused with FIG. 7.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS Background Theory

In order to facilitate a better understanding of the specificembodiments of the present invention, it may be preferable to begin bydiscussing the existing systems of modulation as used with analogsignals. These fall into two basic categories, each with its ownadvantages and disadvantages. Amplitude modulation as used in standardbroadcasting, is a double sideband signal (AM-DSB) system including thecarrier frequency and requires a bandwidth double the audio bandwidthsince there is a sideband on each side of the carrier frequency for eachfrequency in the modulating signal. Suppressing the carrier frequency(AM-DSB-SC) improves the signal-to-noise ratio by as much as 6 dB.Single sideband AM transmission, as used for other than the broadcastband, reduces the required bandwidth by half, is typically used with asuppressed carrier (AM-SSB-SC), and the signal-to-noise ratio is betterthan with AM-DSB or AM-DSB-SC. The second basic method of modulation isangular modulation which may be either frequency (FM) or phase (PM)modulation. In angular modulation, only the frequency (or phase) of thecarrier frequency is varied by the modulating signal. The requiredbandwidth depends on the modulating index and, for normal broadcasting,is at least ten times the audio bandwidth which provides a significantimprovement in signal-to-noise ratio. This may be termed wideband FM,usually referred to merely as FM. In narrowband FM (FM-NB) themodulating index may be as low as 1.0, making the bandwidthapproximately the same as for AM-DSB, but the signal-to-noise ratio isstill greater than for any form of AM, assuming that both AM and FM usepre-emphasis and de-emphasis.

In transform modulation (TM) the advantages of both AM and FM arecombined; i.e., for any given transmission bandwidth, thesignal-to-noise ratio will be considerably greater with TM than with anyform of AM or FM (PM) and the "capture" characteristic of FM isretained. This can best be seen by first recognizing that an FM signalcan be represented by either:

    a=A.sub.c sin [ω.sub.c t+m.sub.f sin ω.sub.a t](1)

where A_(c) is the carrier amplitude, ω_(a) is 2πf_(a), the audiofrequency, and m_(f) is ΔF/f_(a) where ΔF is the peak frequencydeviation, or by

    a=A.sub.c {[cos (m.sub.f sin ω.sub.a t)] sin ω.sub.c t+[sin (m.sub.f sin ω.sub.a t)] cos ω.sub.c t)}      (2)

Equation 2 is thus seen to be a form of double sideband, suppressedcarrier, quadrature modulated AM system with both sidebands required forcoherent decoding of the original input signal. In distinction to knownAM quadrature signals, however, each of the modulating signals, cos(m_(f) sin ω_(a) t) and sin (m_(f) sin ω_(a) t) is a "transform" of theoriginal audio signal multiplied by the modulation index. Thus, each ofthe modulating signals has a limited peak magnitude, limited by themaximum values of the sine and cosine. This amounts to "folding" eachsignal back upon itself. It is this folding process which provides boththe capture characteristic and the higher signal-to-noise ratio of FMsince, when the signal is "unfolded" the level of any noise introducedduring transmission remains unchanged.

In order to provide an ideal signal it is necessary to determine whatfunction of the original signal, when used to modulate the carriersingle sideband, would allow coherent decoding at a receiver. Such afunction must have the following characteristics: It must somehow "fold"the signal back upon itself in order to maintain a constant peakamplitude of the carrier; it must do the "folding" in a manner whichproduces the minimum number of sidebands (narrowest possible bandwidth);and, of course, it must be a function which is possible to obtain and todecode.

As will become apparent, the ideal way to obtain such a function is totransform the signal m_(f) sin ω_(a) t to sin (m_(f) sin ω_(a) t) whichhas a constant peak amplitude for values of m_(f) greater than π/2. Whena single carrier is amplitude modulated with this transform signal(herein termed "TM" for "transform modulation"), and one set ofsidebands is removed before transmission, the signal-to-noise ratio ofthe received signal is the same as an FM signal which the samemodulation index (and twice the bandwidth) and better than an AM signalof the same bandwidth.

FIG. 1 illustrates how an understanding of the equation (2) as givenabove can be used to provide a new transmission system for standard FM.A signal f(t) is coupled from a terminal 10 to two function generators12, 14, which provide the complementary functions f₁ (t) and f₂ (t)which are sin f(t) and cos f(t) which will be referred to hereinbelow.The signals from function generators 12, 14 may be coupled throughlowpass filters 15 which would pass all frequencies necessary to allowrecovery of sin f(t) and cos f(t) with satisfactory accuracy. Thesesignals are coupled as inputs to a form of AM quadrature system 16(shown in dashed line) and are, within the limits of the transmissionsystem, the output signals of the system 16. The system 16 includes anencoding or modulation portion having a frequency source 18 forproviding sin ω_(c) t, a phase shifter 20 for producing a quadraturesignal cos ω_(c) t, a multiplier 22 coupled to function generator 12 andfrequency source 18, a multiplier 24 coupled to function generator 14and phase shifter 20, and an adder 26 coupled to combine the outputs ofmultipliers 22, 24. The adder output is connected by a link 28 to ademodulation portion of the system 16 which may contain a frequencysource 30, a phase shifter 32, a multiplier 34 coupled to the link 28and source 30, a multiplier 36 coupled to the link 28 and the phaseshifter 32. The frequency source 30 may be synchronized with thefrequency source 18. It will be seen that the outputs of the multipliers34, 36, are essentially f₁ (t) and f₂ (t). The FM signal in the link 28can be an RF signal which may amplified, translated in frequency andtransmitted, thus the link 28 may be merely one wire or a completetransmitting/receiving system.

Keeping in mind that the modulating signals were not simply f(t), theinput signal at terminal 10, but the sine and cosine functions of f(t),it will be seen that the signal at the link 28 is the signal of Equation2 above, a constant amplitude, frequency-modulated signal. A decoder 38receives the two signals sin f(t) and cos f(t) and derives f(t). Itwill, of course, be understood that the link 28 could couple to anyother form of standard FM receiver.

FIGS. 2a, 2b and 2c are possible embodiments of the decoder 38 ofFIG. 1. In FIG. 2a, the f₁ (t) signal is coupled to a differentiatingcircuit 40 with output signal [(df(t)/dt)] cos f(t) coupled to a divider42. In the divider, this signal is divided by cos f(t) and the output iscoupled to an integrator 44 with output signal f(t) which is theoriginal input at terminal 10 of FIG. 1.

In FIG. 2b, the output of the differentiator 40 is coupled to one inputof an op amp 46, which functions as a comparator, the output of which iscoupled to the integrator 44. The output of the op amp 46 is alsocoupled back to a multiplier 48. The second input of the multiplier 48is the input signal f₂ (t), and the feedback forces the output of themultiplier, coupled to the op amps, to equal the differentiator output.Thus, the input and output signals of the integrator 44 are as in FIG.2a.

In FIG. 2c, f₂ (t) is coupled to the op amp 46, and the amplifier outputis coupled back to a multiplier 50. The signal f₁ (t) is coupled to asecond input of multiplier 50, and the output is coupled to anotherintegrator 52. The integrator 52 output is thus forced to be f₂ (t) andis coupled to the op amp 46 and again the integrator 44 functions asabove.

In FIG. 3, sin f(t)+any extraneous noise is represented by "Q" and cosf(t)+noise by "I" for simplicity in the drawing figure. As is known, thefrequency modulation on the carrier is the rate of change of carrierphase angle φ or ##EQU1## Deriving this rate-of-change signal usuallyrequires a limiter/discriminator combination. To implement thederivation of the modulating signal f(t), the signal Q is squared in amultiplier 53 and signal I is squared in a multiplier 55. Signals I² andQ² are summed in an adder 56 and the sum is coupled to a divider 58.Signal Q is also coupled to a multiplier 59 and to a differentiator 60,the output of the latter being coupled to a multiplier 62. Signal I isalso coupled to the multiplier 62 and to a differentiator 64, the outputof the latter being coupled to the multiplier 59. The output signals ofmultipliers 59 and 62 are coupled to a subtractor 66 and the subtractoroutput signal is thus ##EQU2## It will be seen that the combination ofmultipliers 59, 62, differentiators 60, 64 and subtractor 66 perform thediscriminator function. This signal is coupled to the divider 58 havingan output of dφ/dt and the integral of that signal, from the integrator44, is f(t), the original signal. It will be seen that the decoder ofFIG. 3 detects only frequency (or phase) modulation and ignoresamplitude variations in the signal caused by noise or interference andhas all the characteristics of other FM systems.

FIG. 4a illustrates a system (TM) providing the S/N ratio and "capture"characteristics of FM systems, including that of FIG. 1, but requiringonly one-half the bandwidth of a comparable FM signal. FIG. 4b shows thecorresponding waveforms for the system of FIG. 4a. The input signal atterminal 68 is f₃ (t) or m_(f) sin ω_(a) t, shown as a sinusoidalwaveform 70, where m_(f) is the modulation index. The output of thefunction generator 12 is now sin [f₃ (t)] or f₄ (t). This latter outputsignal is shown in the waveform 72 and is "folded back" upon itselfwithin the constraints of the sine values ±1. The waveform 72 is themodulating signal coupled to a single sideband (SSB) transmitter 74. Thetransmitted signal may be received by an SSB receiver 76 with extraneousamplitude modulations caused by noise or interference. Therefore, thedemodulated signal 78 may be f₄ (t)+n, n=noise, rather than beingidentical to waveform 72. When the waveform 78 is "unfolded" in afunction generator 80 which essentially derives the arc sine of thegenerator input signal, the output at a terminal 82 is the waveform 84f₃ (t)+n. Thus, as in angular modulation, the original waveform 70 hasbeen restored with only a slight amount of noise, giving asignal-to-noise ratio improvement factor of m_(f), and with theimportant advantage that one signal set of sidebands produces the sameimprovement over AM as does FM with the same modulation index and twicethe bandwidth.

The diagram of FIG. 5 shows many similarities to FIG. 1; i.e., an inputsignal f(t) at the terminal 10 is transformed in the function generator12 to sin f(t), filtered in filter 15, and coupled to the system 16, andthe sin f(t) output of the system 16 is coupled to the decoder 38.However, instead of coupling the signal f(t) to the second functiongenerator 14 (as in FIG. 1), the sin f(t) output from function generator12 is also coupled to a phase-shifting network 88 which is not frequencyselective and which shifts the phase of each audio component by 90°, anoperation known mathematically as taking the Hilbert transform of asignal. The output signal of the network 88 is then coupled to themultiplier 24, making the composite signal at the link 28 as SSB signal.This method of modulating a DSB-SC transmitter with the modulatingsignal on one input terminal and the Hilbert transform of the modulatingsignal on a second terminal is known as the phase-shift method ofgenerating an SSB signal, thus the signal at link 28 is an SSBtransmission of sin f(t) or TM. When the TM signal is demodulated in aquadrature detector, one output is sin f(t) and the other, from themultiplier 36 is the Hilbert transform of sin f(t). The output signal ofthe multiplier 36 is coupled to a phase-inverting switch 90, which iscontrolled by a control signal sgn [df(t)/dt] from a terminal 92,producing an output signal which is approximately cos f(t). The controlsignal will be discussed with respect to FIG. 7. As in FIG. 1, the sinf(t) and cos f(t) signals are processed in the decoder 38 to provide anoutput of f(t). In this circuit it would be desirable, as is known inthe art, to send a small signal at the carrier frequency of source 18through the link 28 for the purpose of synchronizing the source 30 bythe usual phase locked means. This small amount of carrier can also beused for AGC as in AM-SSB.

The waveform of FIG. 6a represents sin f(t) where f(t)=m_(f) sin ω_(a)t, ω_(a) t is a single tone audio frequency and m_(f) is the modulatingindex (10 radians in this example). The "folding" effect of thesine-of-the-sine function is apparent in this figure. FIG. 6b representsthe Hilbert transform of the signal of FIG. 6a. FIG. 6c is the signal atterminal 92 and may be expressed as sgn [df(t)/dt], representing ±1,depending on the sign of the given function. While FIG. 6d isapproximately cos f(t), the more accurate cos f(t) signal of FIG. 6e isto be preferred, and a means of deriving FIG. 6e will be shown in anddescribed with respect to FIG. 8.

In FIG. 7, a terminal 94 couples the signal sin f(t) from the multiplier34 to the decoder 38. A terminal 96 couples the Hilbert transform signalfrom multiplier 36 to the phase-inverting switch 90 and also to a zerocrossing detector 97 comprising a limiter 98, a differentiator 100 and afull wave rectifier 102. The detector 97 detects the zero crossings ofthe signal at the terminal 96, and is coupled to one input of an ANDgate 104. A terminal 106 of the decoder 38 provides a signal df(t)/dt toa full wave rectifier 108. The rectified output is coupled to adifferentiator 110 and to an adder 112 which has a second input from thedifferentiator 110. The combined output of the adder 112 thus includessmall positive pulses which precede the zero crossings of the signalfrom the terminal 106. These positive pulses trigger a one-shotmultivibrator 114, the output of which is coupled to a second input ofthe AND gate 104. The output of the AND gate then represents only thosezero crossings in the output of the detector 97 which coincide closelywith the zero crossings of the signal at terminal 106. The coincidentcrossings are shown within the windows 116 indicated in line B of FIG.6. The AND gate output triggers a flip-flop 118 which provides thesignal of line C of FIG. 6.

In FIG. 8, a coherence circuit 120 has been added between thephase-inverting switch 90 and the decoder 38 (FIG. 7) to prevent theoutput signal of the switch 90 from approaching zero at the same timethe signal at terminal 94 goes to zero, since this event would cause thedecoder 38 to lose coherence and produce a large burst of noise. Anoutput terminal 122 of decoder 38 provides a signal (I² +Q²) to acomparator 124 having a reference terminal 126. The comparator output iscoupled to a phase-inverting switch 128 which is controlled by a signal(sgn [cos f(t)]) from a F/F 130. The signal from the phase-invertingswitch 90 is combined with the output of the phase-inverting switch 128in an adder 132, the output of which is essentially cos f(t) as seen inFIG. 6e. The adder output is coupled to decoder 38 and, through alimiter 134 and a differentiator 136, to the F/F 130, providing a signalsgn [cos f(t)] which controls the switch 128. When the signals atterminals 94 and 96 both approach zero, the comparator 124, through theswitch 128, provides a signal which brings the cos f(t) signal to theproper polarity. In this circuit the small carrier signal referred towith respect to FIG. 5 can also be rectified and amplified to providethe reference voltage at terminal 126 of the comparator 124. With thistype of reference voltage, the receivier can determine when there is notransmission, and squelch as necessary.

Thus, there has been shown and described a new system of modulationwhich can provide all of the desirable characteristics of both AM and FMwith minimum bandwidth requirements, and a new way of providing standardFM. A cosine signal correction circuit for use in such a system has beendisclosed and claimed. An improved circuit for deriving the transformsignal which is not transmitted has been shown, with the correctioncircuit increasing the accuracy of the derived signal. Othermodifications and variations are possible, and it is intended to coverall such as fall within the spirit and scope of the appended claims.

What is claimed is:
 1. A cosine signal corrector comprising:first inputmeans for receiving a first input signal which is approximately cos f(t)where f(t) is an information signal; second input means for receiving asecond input signal which is approximately equal to sin² f(t)+cos² f(t);third input means for providing a reference signal; comparator meanscoupled to the second and third input means for providing an errorsignal in response to any error in the approximated second input signal;adder means coupled to an output of the comparator means and to thefirst input means for adding the error signal to the approximate cosf(t) signal; and circuit means coupled to an output of the adder meansfor deriving sgn cos f(t), and coupled to the output of the comparatormeans for inverting the phase of the error signal in response to sgn cosf(t).
 2. A cosine signal corrector according to claim 1 and furtherincluding fourth input means for receiving a signal which is essentiallythe Hilbert transform of sin f(t), phase inverting means coupled to thefourth input means and means for providing a signal sgn (df(t)/dt) forcontrolling the phase inverting means, the phase inverting meansproviding the first input signal.
 3. A cosine signal corrector accordingto claim 1 and wherein the circuit means comprises limiter means coupledto the adder means output, differentiating means coupled to the limitermeans output, bistable means coupled to the differentiating meansoutput, and phase inverting means coupled to control the comparatormeans output signal in response to the output signal of the bistablemeans.